The classical Douglas formula [1]} (see also Radó [2]} and Chen and Fukushima [3]})
relates the energy of the harmonic function \(u\)  on the unit disk \(B(0,1)\subset \mathbb {R}^2\)  to the energy of its boundary values \(g\)  on the boundary of the disk, identified with the torus \([0,2\pi )\) :
\(\int \limits _{B(0,1)}|\nabla u(x)|^2\, dx = \frac{1}{8\pi }\int _0^{2\pi }\int _0^{2\pi }\frac{(g(\eta ) - g(\xi ))^2}{\sin ^2((\xi - \eta )/2)}\, d\eta \, d\xi .\) 
